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The Crab and Vela are well-studied glitching pulsars and the data obtained so far should enable us to test the reliability of models of their internal structures. Very recently it was proposed that glitching pulsars are embedded in bimetric spacetime: their incompressible superfluid cores (SuSu-cores) are embedded in flat spacetime, whereas the ambient compressible and dissipative media are enclosed in Schwarzschild spacetime. In this letter we apply this model to the Crab and Vela pulsars and show that a newly born pulsar initially of
and an embryonic SuSu-core of

*and*

*, respectively. Furthermore, the under- and overshootings phenomena observed to accompany the glitch events of the Vela pulsar are rather a common phenomenon of glitching pulsars that can be well-explained within the framework of bimetric spacetime.*

The Crab and Vela pulsars are well-known and extensively studied pulsars (see [

In [

The strategy of obtaining the optimal values here relies on using a global iterative solution procedure that takes the following constraints into account (see also

1) The elements of the sequence { Δ Ω g Ω } n must fulfill the three conditions:

l { Δ Ω g Ω } n → n → 0 0 , which means that the media in both the core and in the surrounding shell must have identical rotational frequency initially.

l Δ Ω g Ω | n = N 0 = Δ Ω g Ω | C r a b = 4 × 10 − 9 and Δ Ω g Ω | n = N 1 = Δ Ω g Ω | V e l a = 2.33 × 10 − 6 , i.e. the elements number N 0 and N 1 ( ≫ N 0 ) of the sequence { Δ Ω g Ω } n must be identical to the observed values of the Crab and to the Vela pulsars, respectively.

Crab | Vela | |
---|---|---|

Mass ( M ⊙ ) | 1.4 | 1.8 |

Age (kyr) | 1.24 | 11.3 |

B (10^{12} G) | 4.875 | 4.35 |

Ω ( s − 1 ) | 200 | 70 |

Δ Ω g / Ω | 4 × 10^{−9} | 2.338 × 10^{−6} |

Δ t g ( yr ) | 1.6 | 2.5 |

l { Δ Ω g Ω } n → n → ∞ α ∞ < ∞ , i.e. the sequence must converge to a finite value. Moreover, our test calculations have shown that for t ≫ t V e l a , ∂ ∂ t ( { Δ Ω g Ω } n ) < 0 as otherwise the magnetic field would fail to spin-down the crust and therefore to surpass Δ Ω c r required for triggering a prompt spin-down of the core into the next lower energy state.

Indeed, one possible sequence which fulfills the above-mentioned constraints, though it might not be unique, is shown in

2) The initial conditions used here are Ω 0 = Ω ( t = 0 ) = 1440 Hz (see [

3) The elements of the sequence { Ω a m } n are obtained through the energy balance equation:

d d t ( 1 Ω a m 2 ) = − α E M B 2 I a m , (1)

where I a m is the inertia of the ambient compressible dissipative medium, which, due to the increase of the SuSu-core, must decrease on the cosmic time and α E M = 4.9 × 10 − 4 is a non-dimensional constant.

Thus, the time-evolution of the core’s rotational frequency proceeds as follows: for a given Ω c n , the ambient medium is set to decrease its frequency continuously with time through the emission magnetic dipole radiation. This implies that the difference Δ Ω c − a m should increase with time until Δ Ω c − a m has surpassed the critical value ≥ Δ Ω c r , In this case three events are expected to occur promptly:

l The rigid-body rotating core changes abruptly its rotational state from E c n , r o t = 1 2 I c n ( Ω c n ) 2 into the next the quantum-mechanically permitted lower energy state: E c n + 1, r o t = 1 2 I c n + 1 ( Ω c n + 1 ) 2 . This process is associated with ejection of a certain number of vortices into the boundary layer (BL) between the core and the overlying dissipative medium.

l The ejected vortices by the core are then absorbed by the differentially rotating dissipative medium and re-distributed viscously. Hence the medium in the BL would experience the prompt spin-up: Ω a m n → Ω a m n + Δ Ω g n , where Δ Ω g n is deduced from the sequence { Δ Ω g Ω } n .

l The radius of the core is set to increases as dictated by the Onsager-feynmann equation:

∮ V ⋅ d l = 2 π ℏ m N , (2)

where V , l , ℏ , m , N denote the velocity vector, the vector of line-element, the reduced Planck constant, mass of the superfluid particle pair and the number of vortices, respectively (see [

( Ω S ) c n + 1 = ( Ω S ) c n ⇒ S n + 1 = ( Ω c n Ω c n + 1 ) S n (3)

where S c n ≐ π ( R c n ) 2 and R c n correspond to the cross-sectional area of the SuSu-core and to the corresponding radius, respectively. The increase in the dimension of the core implies that the matter in the geometrically thin boundary layer between the SuSu-core and the ambient medium should undergo a crossover phase transition into an incompressible superfluid, whose total energy density saturates around the critical value ρ c r ≈ 6 ρ 0 (see [

The set of equations consists of the TOV equation for modeling the compressible dissipative matter in the shell overlaying the incompressible gluon-quark superfluid core, whereas the latter is set to obey the zero-torque condition and to dynamically evolve according to the Onsager-Feymann equation (for further details see Sec. 2 and Eq. 10 in [

The global iteration loop is designed here to find the optimal values of the parameters: α 0 , α 1 , the elements of the sequence Δ Ω c r n and the decay rate of the magnetic field. These values should fulfill the initial and final conditions, the currently observed values of the time passages between two successive glitch-events Δ t g both of the Crab and the Vela pulsars, the current observed values of their magnetic fields masses.

Indeed, our intensive computations reveal that optimal fitting may be achieved for M c ( t = 0 ) ≈ 0.029 M ⊙ , a sequence of Δ Ω g n , whose elements are shown in

Due to the incompressible, superfluid and supreconducting character of the core, the evolution of the magnetic field is solely connected to the dynamics of the ambient compressible and dissipative matter in the shell as well as to its dimensions (see [

may clarify the very weak decay of magnetic field as pulsars evolve from the Crab to the Vela phase. Mathematically, let the magnetic energy in a shell of a newly born pulsar be:

E M = ∫ B 2 8 π d v ~ B 2 6 ( R ⋆ 3 − R c 3 ) , (4)

where R ⋆ denotes the pulsar’s radius. Assuming E M to roughly decay as the rotational energy E Ω , then we obtain:

B − = α Ω Ω a m 2 M a m 1 / 2 R ⋆ 2 − R c 2 R ⋆ 3 − R c 3 , (5)

where α Ω is constant coefficient.

On the other hand, dynamo action in combination with magnetic flux conservation and other enhancement mechanisms would contribute positively to the magnetic field, that can, for simplicity absorbed in the term: B + = α B / ( R ⋆ 2 − R c 2 ) . The coefficient α B is set to ensure that the magnetic field remains in the very sub-equipartition regime. Hence the interplay between magnetic loss and enhancement would yield an effective magnetic field that evolves according to:

B t o t = α B R ⋆ 2 − R c 2 − α Ω Ω a m 2 M a m 1 / 2 R ⋆ 2 − R c 2 R ⋆ 3 − R c 3 . (6)

Consequently, our model predicts that the decreasing volume of the shell enclosing the ambient medium in combination with dynamo action in the boundary layer could potentially be the mechanism that keeps the decay of magnetic fields in pulsars extremely weak.

In fact our model predicts the glitch activity of a newly born pulsar, which evolves into a Crab phase, followed by a Vela phase and finally by an invisible phase, to be approximately two orders of magnitude larger than it was estimated by other models (see

model pulsar may undergo millions or up to billions of glitches during their luminous life time with passages of time between two successive glitch events that range from nanoseconds in the very early time up to hundreds or even thousands of years toward the end of their luminous life times (see

Moreover, the model also predicts the occurrence of under- and overshootings that have been observed to accompany the glitch events in the Vela pulsar (see [^{1}, this enhancement is communicated to the crust via Alfven waves, V A , whereas the excess of rotational energy is communicated via shear viscosity with an effective propagational velocity V v i s . As these two speeds are generally different with V A > V v i s in most cases, the time-delay in the arrival of communication enforces the crust to react differently. Specifically, the arrival of magnetic enhancement prior to the rotational one leaves the crust subject to an enhanced magnetic braking and therefore to a stronger reduction of its rotational frequency (see the top panel of

Indeed, in the case of Vela, the propagational speed of Alfven waves may be esitmated to be of order V A ~ B / ρ ≈ 10 8 cm / s . Hence the enhanced MFs in the BL would be communicate to the crust within δ τ M F = Δ R / V A = ( R ⋆ − R c ) / V A ≈ 10 − 2 s . On the other hand, supplying the crust with rotational energy would proceed on the viscous time scale, which is estimated to be: δ τ v i s = ( Δ R ) 2 / ν v i s (see [

*I.e. In the absence of magnetic monopoles.

Moreover, the observed order in which undershooting followed by overshooting is an indication for a time-delay in the arrival of communication resulting from V A > V v i s and from the significant difference of the locations of the BL and the crust. This order is expected to reverse if V A < V v i s .

In fact, the under- and overshooting here may indicate that MFs are insensitive to the momentary rotational frequency of the crust, but rather to the activity and dynamics of the matter in the BL.

Extending this analysis to both the Crab and Vela pulsars, the relative time-delays is expected to be: δ τ v i s C r a b / δ τ v i s V e l a ~ ( Δ R C r a b / Δ R V e l a ) 2 ≈ 3.4 or equivalently, the undershooting in the case of the Crab is expected to last 3.4 sec comapred to one second in the Vela case.

Finally, although the physics is entirely different, the situation here is strikingly similar to action of the solar dynamo, which is considered to be located in the so-called tachcline between the rigid-body rotating core and the overlying convection zone [

The calculations have been carried out using the computer cluster of the IWR, University of Heidelberg. RS acknowledges the use of KAUS baseline research funds

The authors declare no conflicts of interest regarding the publication of this paper.

Hujeirat, A. A. and Samtaney, R. (2020) How Massive Are the Superfluid Cores in the Crab and Vela Pulsars and Why Their Glitch-Events Are Accompanied with under and Overshootings? Journal of Modern Physics, 11, 395-406. https://doi.org/10.4236/jmp.2020.113025